Closure and interior in a three-point chain topology

Goal and scope

Find the closure and interior of A={b}.

An exhaustive finite-topology calculation; no metric is supplied.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, the entire candidate family is tau={empty,{a},{a,b},X}, and A={b}.

• \tau — complete open family
• \overline A — closure
• A^\circ — interior

Rules and sources

• mko_definition · cited — First validate the complete finite topology, then calculate closure and interior exactly from all closed supersets or open subsets. Retain the empty-set and whole-space boundaries and dependence on the topology; do not silently introduce a metric. · mko-closure-and-interior
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks.
• definition_here · prerequisite — Closed sets are relative complements of opens; closure intersects all closed supersets and interior unites all open subsets.

Main derivation

\tau=\{\varnothing,\{a\},\{a,b\},X\}

\varnothing,X\in\tau

\varnothing\subset\{a\}\subset\{a,b\}\subset X

U\cup V=\max_{\subseteq}(U,V),\quad U\cap V=\min_{\subseteq}(U,V)

F=X\setminus U\ (U\in\tau)

\mathcal F=\{X,\{b,c\},\{c\},\varnothing\}

A=\{b\},\quad A\subseteq F\in\mathcal F

\{F\in\mathcal F:A\subseteq F\}=\{\{b,c\},X\}

\overline A=\bigcap\{F\text{ closed}:A\subseteq F\}

\overline A=\{b,c\}\cap X=\{b,c\}

U\in\tau,\quad U\subseteq\{b\}

\{U\in\tau:U\subseteq A\}=\{\varnothing\}

A^\circ=\bigcup\{U\in\tau:U\subseteq A\}

A^\circ=\varnothing

A^\circ\subseteq A\subseteq\overline A

\varnothing\subseteq\{b\}\subseteq\{b,c\}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S05, S07 — The inclusions are consistency checks; the exhaustive intersection and union establish the actual constructions.
• S07 · premise_audit · S06 — Empty is open and contained in A, and no larger open subset was omitted.
• S06 · inverse_equivalence · G1, S02 — All three nonempty opens have been checked, rather than inferred from a label.
• S05 · premise_audit · S04 — The result is closed and contains A; every closed superset contains it, verifying minimality.
• S04 · inverse_equivalence · G1, S03 — Both selected sets contain b and both rejected sets do not; all four have been exhausted.
• S03 · premise_audit · G1, S02 — Complementing the four closed sets returns exactly the topology and checks the ambient carrier.
• S02 · premise_audit · G1, S01 — The chain argument covers the complete listed family, not one tested pair.
• S01 · premise_audit · G1 — Endpoint membership alone is insufficient; intersections and unions still need checking.

Error controls and negative controls

Whole-family obligation

List every closed superset before intersecting and every open subset before uniting.

Ambient binding

All complements are relative to X, not A.

• The closure is X because X is closed and contains A. → reject — The smaller closed superset {b,c} was omitted.
• The interior of {b} is {b}. → reject — {b} is absent from the complete open family.
• These answers hold for every topology on X. → reject — In the discrete topology {b} itself is open and closed.

Result and limitations

In this topology the closure is {b,c} and the interior is empty. Empty and X each equal their own closure and interior.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 4d2edbe08895d69e94f4ee3bf38a0402fcb0ae2147791f8a665f438b55ac9303
• original source SHA-256: 2fa41873c0ddcc9253a540235a90fa71fd13d0554b298df20b082b0267e4e54a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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